New fixed point theorem and its application to ODE

TL;DR

This paper introduces a novel fixed point theorem that merges contraction principles with order-theoretic concepts, enabling the proof of existence for solutions to ODEs with discontinuous vector fields, extending classical results.

Contribution

A new fixed point theorem combining contraction and order-theoretic methods, leading to generalized existence results for discontinuous ODEs.

Findings

Established a fixed point theorem unifying contraction and Knaster-Tarski principles.

Proved an existence theorem for ODEs with discontinuous vector fields.

Generalized Carathéodory's theorem for broader classes of differential equations.

Abstract

We prove a fixed point theorem that combines the contraction mapping principle and some Knaster-Tarski-like theorem. As a consequence we obtain an existence theorem to initial value problem for ordinary differential equation with discontinuous vector field. This theorem generalizes Carath\'eodory's existence theorem.